Viscous fingering and the shape of an electronic droplet in the quantum Hall regime.

We show that the semiclassical dynamics of an electronic droplet, confined in a plane in a quantizing inhomogeneous magnetic field in the regime where the electrostatic interaction is negligible, is similar to viscous (Saffman-Taylor) fingering on the interface between two fluids with different viscosities confined in a Hele-Shaw cell. Both phenomena are described by the same equations with scales differing by a factor of up to 10(-9). We also report the quasiclassical wave function of the droplet in an inhomogeneous magnetic field.

Integrable structure of interface dynamics

We establish the equivalence of 2D contour dynamics to the dispersionless limit of the integrable Toda hierarchy constrained by a string equation. Remarkably, the same hierarchy underlies 2D quantum gravity.